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2014 Issue 445

  1. Prove that \[\overline{\underset{2014\text{ digits}}{\underbrace{111\ldots111}}\underset{2014\text{ digits}}{\underbrace{222\ldots222}}}-\overline{\underset{2014\text{ digits}}{\underbrace{333\ldots333}}}\] is a perfect square.
  2. Given a triangle $ABC$ with $\widehat{BAC}>90^{0}$ and the lengths of its sides are three consecutive even numbers. Find these lengths. 
  3. Let $a,b$ be two positive real numbers such that $a+b$, $ab$ are positive integers and $[a^{2}+ab]+[b^{2}+ab]$ is a perfect square, where $[x]$ is the greatest integer not exceeding $x$. Prove that $a,b$ are positive integers.
  4. Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$. On the opposite rays of the rays $DA$, $EB$, $FC$ choose three points $M,N,P$ respectively such that $\widehat{BMC}=\widehat{CNA}=\widehat{APB}=90^{0}$. Prove that the lines containing the sides of the hexagon $APBMCN$ are both tangent to a circle.
  5. Find all integers $m$ such that the equation \[x^{3}+(m+1)x^{2}-(2m-1)x-(2m^{2}+m+4)=0\] has an integer solution.
  6. Given any triple of real numbers $a,b,c>1$. Prove the following inequality \[(\log_{b}a+\log_{c}a-1)(\log_{c}b+\log_{a}b-1)(\log_{a}c+\log_{b}c-1)\leq1.\]
  7. Let $ABC$ ($AB<AC$) be an acute triangle inscribed in a circle $(O)$. The altitudes $AD$, $BE$, $CF$ intersect at $H$. Let $K$ be the midpoint of $BC$. The tangent lines to the circle $(O)$ at $B$ and $C$ meets at $J$. Prove that $HK$, $JD$, $EF$ are concurrent.
  8. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is bounded on a certain interval containing $0$ and $f$ satisfies \[2f(2x)=x+f(x)\] for every $x\in\mathbb{R}$.
  9. Let \[f(x)=x^{3}-3x^{2}+9x+1964\] be a polynomial. Prove that there exists an integer $a$ such that $f(a)$ is divisible by $3^{2014}$. 
  10. Does there exist a continuous funtion $f:\mathbb{R}\to\mathbb{R}$ satisfying the following property: for any $x\in\mathbb{R}$, among $f(x)$, $f(x+1)$, $f(x+2)$ there are exactly two rational numbers and one irrational number?.
  11. Given a sequence $\{a_{n}\}_{1}^{\infty}$ where \[a_{1}=1,\,a_{2}=2014,\quad a_{n+1}=\frac{2013a_{n}}{n}+\left(1+\frac{2013}{n-1}\right)a_{n-1}.\] Find \[\lim_{n\to\infty}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}\right). \]
  12. Let $ABCD$ be a quadrilateral circumscribing a circle $(I)$. The sides $AB$ and $BC$ are tangent to $(I)$ at $M$ and $N$ respectively. Let $E$ be the intersection of $AC$ and $MN$, and $F$ be the intersection of $BC$ and $DE$. $DM$ intersects $(I)$ at another point, say $T$. Prove that $FT$ is tangent to $(I)$.

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Mathematics & Youth: 2014 Issue 445
2014 Issue 445
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_93.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_93.html
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